Utility of Phenomenological Models for Describing Temperature Dependence of Bacterial Growth

Transcription

1 APPLIED AND ENVIRONMENTAL MICROBIOLOGY, Sept. 1991, p /91/ $2./ Copyright C 1991, American Society for Microbiology Vol. 57, No. 9 Utility of Phenomenological Models for Describing Temperature Dependence of Bacterial Growth ARMIN HEITZER,t HANS-PETER E. KOHLER, PETER REICHERT, AND GEOFFREY HAMER* Swiss Federal Institute of Technology Zurich and Swiss Federal Institute for Water Resources and Water Pollution Control, CH-86 Duibendorf, Switzerland Received 28 March 1991/Accepted 28 June 1991 We compared three unstructured mathematical models, the master reaction, the square root, and the damage/repair models, for describing the relationship between temperature and the specific growth rates of bacteria. The models were evaluated on the basis of several criteria: applicability, ease of use, simple interpretation of model parameters, problem-free determination of model parameters, statistical evaluation of goodness of fit (X2 test), and biological relevance. Best-fit parameters for the master reaction model could be obtained by using two consecutive nonlinear least-square fits. The damage/repair model proved to be unsuited for the data sets considered and was judged markedly overparameterized. The square root model allowed nonproblematical parameter estimation by a nonlinear least-square procedure and, together with the master reaction model, was able to describe the temperature dependence of the specific growth rates of Klebsiella pneumoniae NCIB 418, Escherichia coli NC3, Bacillus sp. strain NCIB 12522, and the thermotolerant coccobacillus strain NA17. The square root and master reaction models were judged to be equally valid and superior to the damage/repair model, even though the square root model is devoid of a conceptual basis. Temperature is an important factor affecting the growth of bacteria. Most bacteria grow over a temperature range of some 3 to 4 C, irrespective of whether they are psychrophiles, mesophiles, or thermophiles (28). Within the growthpermissible temperature range for any specific strain, four subranges can be identified as the temperature is increased from the minimum to the maximum for growth. These subranges are described as the sub-arrhenius, Arrhenius, optimum, and superoptimum ranges, respectively. In addition, temperatures below the minimum growth temperature and those above the maximum growth temperature constitute the subminimum and supermaximum ranges, respectively. Within the growth-permissible temperature range, temperature affects the specific growth rate and changes in temperature will influence both the metabolism and performance of bacteria (17, 18, 2). As far as bacterially mediated engineered processes are concerned, the Arrhenius, optimum, superoptimum, and supermaximum temperature subranges are of the greatest interest, whereas in most natural environments, bacteria are subjected to the subminimum, the sub-arrhenius, and the Arrhenius subranges. Several unsegregated and unstructured phenomenological models have been proposed to describe the growth of bacteria at temperatures throughout the entire growth-permissible temperature range. Provided that such models effectively describe the macroscopic growth characteristics of diverse bacterial cultures at various temperatures and provided that their limitations, primarily the lack of insight they provide with respect to either microscopic or molecular mechanisms, are fully recognized, they are valuable guides for process design and operation and the prediction of culture performance and growth behavior from limited data. In some respects, phenomenological models are to biology * Corresponding author. t Present address: The University of Tennessee Center for Environmental Biotechnology, 1515 Research Drive, Knoxville, TN what classical thermodynamics has been to chemistry. Their value is in giving a macroscopic description rather than a microscopic understanding of particular processes. In the present contribution, the objective is to statistically evaluate the fidelity and determine the utility of three different phenomenological models that have been proposed for describing growth over the entire permissible temperature range of any particular microbial strain. The judgment criteria for the utility of the three models are their ease of use, their applicability to any complete range of growth temperatures, their ability to maintain a high degree of biological reality, simple interpretation of their respective parameters, and problem-free determination of these parameters. To this end, five independently determined data sets have been used as the basis for evaluation. THEORY The earliest descriptions of the relationship between biological process rates and temperature (5, 8) were based on the Arrhenius equation, i.e., k = A exp (-EA/RT) (1) in which k is the reaction rate constant, A is the Arrhenius frequency factor, EA is a temperature characteristic or apparent activation energy, R is the gas constant and T is the absolute temperature. Equation 1 was derived by analogy with the van't Hoff equation, which expresses the change with temperature of an equilibrium constant. Subsequently (1, 13), the temperature dependence of the reaction rate constant was also expressed as kbkt k = exp (ASIR) exp (-AH/RT) h (2) by using Eyring's transition state theory as a basis. According to equation 2, in which kb is Boltzmann's constant, h is Planck's constant, and K is the transmission coefficient, the rate constant can be related to the quasithermodynamic properties of the activated complex, i.e., to both &H, the

2 VOL. 57, 1991 enthalpy of formation of the activated complex, and AS', the entropy of formation of the activated complex. Examination of the temperature dependence of the logarithm of the rate constants in both equations 1 and 2 reveals that ( kbkte A h exp (ASIR) Equation 3 shows that the Arrhenius frequency factor, A, contains factors for the frequency of vibration in the reaction coordinate, kb T/h, and for orientation, exp (ASIR) (1). When either equation 1 or 2 was used to describe bacterial growth at various temperatures, neither was found to be applicable for the whole of the growth-permissible temperature range. As a result, encompassing models, which sought to account for the several significant deviations from Arrhenius-type behavior that occur, were developed. Master reaction models. Assuming that growth is the end product of a sequence of enzymatic reactions and that only one reaction in this sequence limits the specific growth rate, a successful description of the substrate dependence of the specific growth rate has been published (26). Therefore, it has been postulated that the effects of temperature on the specific growth rate can be described in a similar manner (7, 11, 13, 21, 32, 33). In such an approach, a rate-limiting master enzyme is considered to be in an equilibrium state between one active and two inactive states resulting either from high- or low-temperature denaturation so that the total enzyme present, [ET], is the sum of the three component states, whereby [ET] is the intrinsic biotic concentration of the rate-limiting enzyme (kilograms of enzyme per kilogram of biomass). When the single enzymatic reaction limits growth under substrate-limiting conditions, the maximum specific growth rate, Pm, can be expressed as a product of the intrinsic concentration of the activated enzyme and the rate constant, k: [ET] 4 1im + L + KH k (4) where KL and KH are the equilibrium constants for low- and high-temperature enzyme inactivation, respectively. After incorporation of equation 2 to account for temperature dependence and substitution of the two equilibrium constants, KH and KL, the following expression for the specific growth rate is obtained: ALm = [ET] kbk [ Air h R RT -ASL AHL-IHrSH AHo] 1 + exp R R + exp +ep[r -RT 1R RTJ (5) According to Sharpe and DeMichele (33), equation 5 is suitable for describing the temperature dependence of the maximum specific growth rate for any poikilothermal organism. In the case when only high-temperature deactivation is considered, equation 5 results in the model proposed by Johnson and Levin (21). Incorporation of equation 1 into equation 4 and consideration of only high-temperature deactivation gives the model proposed by Esener et al. (11). To evaluate the validity of equation 5, it was necessary to transform certain parameters in accordance with the proposals of Schoolfield et al. (32), so as to make the equation suitable for the application of nonlinear least-square techniques. The transformations involved setting (3) and TEMPERATURE/GROWTH MODEL UTILITY 2657 [ET] kbk /AS r(25c) = h K exp R hr TO.5L = AHL /ASL TO.SH = AHH IASH so that equation 4 becomes ILm = T Ke [ 1 (2) K P R t K R K R K} 1) AHTL 1 1] RAH /.1 T)] 1+exp~~I +±exp R \TO5L Tj R \TO5H T] where r(25.c) is the specific growth rate at 25 C ( K) when no master enzyme inactivation occurs, To 5H is the temperature at which the same amount of master enzyme is active and high-temperature inactive, and TO.5L is the temperature at which the same amount of master enzyme is active and low-temperature inactive. If only high-temperature deactivation is considered, equation 6 reduces to T l1 1\ r(25c) Kexp R ( K -T =^- [Rh R (TAH TO.5H T(7 The structure of equation 6 is such that a bell-shaped curve, which asymptotically approaches the abscissa at both its limits, results for the relationship between the maximum specific growth rate and temperature. Damage/repair model. In an alternative approach to modeling the effects of temperature on the maximum specific growth rate of microbes, Fiolitakis et al. (14) have proposed that, in addition to the conventional Arrhenius approach, damage and repair processes should be taken into account. Therefore, equation 1 was extended by the introduction of an activity state variable, V, so that (6) ( EA m = Vexp Ko- RT) (8) where exp (KO) is the Arrhenius frequency factor, EA is the activation energy, and T is the absolute temperature. The activity state variable in equation 8 depends on the rate of damage, cu, which, expressed in dimensionless form, is 1 (9) 1 + exp [a3-a2 (T K)] ' ' where a2 is the damage parameter describing temperature sensitivity and a3 is the damage parameter corresponding to the temperature inflection point. To account for repair, the repair rate, p, was expressed in terms of the activity state variable as p = Vexp (-IP2V) (1) It is assumed that microorganisms compensate for higher damage rates by lowering the activity state variable, such that at any steady state the condition a = p is valid. The boundary conditions are (i) if or ' Pmax with Pmax = 1/(e2), then V =, and (ii) if or ' Pmin with Pmin = 133 exp ( )'

3 2658 HEITZER ET AL. then V = P3 = constant; under these conditions, only the descending branch of the curve resulting from equation 1 is considered. The curve resulting from equation 8 with defined boundary conditions approaches the abscissa asymptotically at suboptimum temperatures, whereas as the maximum growth temperature is approached, a point of no return in terms of repair occurs such that the curve intersects the abscissa and the specific growth rate reaches zero discontinuously. Square root model. In a third, distinctly different approach, Ratkowsky et al. (31) have proposed that a square root relationship exists between the maximum specific growth rate and the absolute temperature so that \~m = b (T- Tmin) (11) where b represents a regression coefficient, T is the absolute temperature, and Tmin is a theoretical minimum temperature for growth. However, to extend the application of equation 11 to the whole of the growth-permissible temperature range, Ratkowsky et al. (3) subsequently extended it so that \/tm = b(t - Tmin){1 - exp [c (T - Tmax)]} where c is an additional regression coefficient and Tmax is a theoretical maximum temperature for growth. To circumvent problems of propagation of errors of the original data, Pm, Kohler et al. (22) have suggested that p,m should be assumed to be a dependent variable, which leads to Rm = b2 (T - Tmin)2{1 - exp [c (T - Tmax)]}2 (12) Additionally, the boundary conditions were given as (i) if T < Tmin, then A.m =, and (ii) if T > Tmax, then un1 Equation 12 and the boundary conditions define =. a composite function which is differentiable over its whole domain. McMeekin et al. (23) have pointed out that a link exists between the square root model and the Arrhenius model and that the square root model is, in fact, a special case of the early growth-temperature relationship developed by Belehradec (6). Fitting of the data. Best-fit curves for the master reaction and square root models with respect to the experimental data were obtained by nonlinear regression using the Marquardt- Levenberg algorithm as implemented by IGOR (WaveMetrics, Lake Oswego, Oreg.). The Marquardt-Levenberg algorithm supplies a covariance matrix, from which estimates for the standard deviations, s, of the model parameters can be obtained (29). These values are given in Tables 1 and 4 for all the fitted parameters. The values for V in the damage/repair model were obtained by solving the equation cr = p, using an algorithm that combines linear interpolation, inverse quadratic interpolation, and bisection (program ZBREN/ DZBREN; IMSL Inc. Math/library, Houston, Tex.). Best-fit curves for the experimental data were then obtained by applying the polytope algorithm (program UMPOL/ DUMPOL; IMSL Inc. Math/library). The standard deviation, s, for the determination of the specific growth rate was.52 h-1 as estimated from four independent measurements for Klebsiella pneumoniae. This value was considered to be representative for the method of determination and was also used for the literature data for which no standard deviations were given. The standard APPL. ENVIRON. MICROBIOL. deviation was also considered to be constant over the whole range of specific growth rate values (homoscedastic data). The error bars in the figures indicate the 95% confidence interval for the corresponding values. As the different model equations are nonlinear in their parameters, initial parameter estimates are required for the fitting methods used. All six parameters of the master reaction model have graphical interpretations, and initial estimates can be obtained as described previously (32). Initial estimates of parameters in the square root model can be calculated as described by Ratkowsky et al. (3). Unfortunately, for the damage/repair model, a trial-and-error procedure is required to get initial parameter estimates. x2 statistic. The probability distribution of x2 can be used to estimate the goodness of fit of a model. In particular, Q, the computed probability that x2 should exceed a particular value by chance, gives a quantitative measure for the goodness of fit of the model. If Q is a very small probability for some particular data set, the apparent discrepancies are unlikely to be chance fluctuations. It is much more likely that the model is wrong or that the measurement errors are markedly greater than stated or are not normally distributed. If Q is too large, too close to unity, literally too good to be true, the cause almost always is that the measurement errors were overestimated (29). In general, a x2 value for a moderately good fit is x2 v, where v, the number of degrees of freedom, equals N - M. N is the number of datum points, and M is number of parameters to be fitted. Q was computed by using an incomplete gamma function (gammaq; IGOR). However, to use the x2 statistic, the standard deviations of the datum points must be taken into account, thus requiring the fits to be weighted. It is also important to note that the standard deviations should be assessed independently from the experimental data to be tested (9). As the usual F test for regression and lack of fit is not generally valid in the nonlinear case, the models were judged on the basis of x2. MATERIALS AND METHODS Data concerning the effect of temperature on the maximum specific growth rate for four different bacteria were evaluated. The bacteria were K. pneumoniae NCIB 418, the thermotolerant gram-negative coccobacillus strain NA17 (2), Escherichia coli NC3, and the thermotolerant methylotrophic Bacillus sp. strain NCIB Data for the first two organisms were generated especially for the present evaluation, and data for the second two were taken from Herendeen et al. (19) and Al-Awadhi et al. (3, 4), respectively. For K. pneumoniae, two different growth media giving significantly different specific growth rate-versus-temperature data were used. Growth media. K. pneumoniae was grown in the mineral salts medium of Evans et al. (12), modified by replacing citric acid with 55 mg of Na2EDTA per liter and supplemented with either 1 g of glucose per liter (mineral medium) or.5 g of glucose per liter, 2.5 g of yeast extract per liter, and 5 g of tryptic soy broth per liter (complex medium) as appropriate. The media were buffered at ph 6.8 with a.1 M Na2HPO4- KH2PO4 mixture. The thermotolerant coccobacillus strain NA17 was grown in the mineral medium described by Al-Awadhi (2), supplemented with 1 g of ethanol per liter as a carbon energy source. Culture methods. K. pneumoniae was grown in 2 ml of medium in magnetically stirred, thermostatted flasks. The temperatures used for growth in mineral medium were 2., 25.1, 3., 35.5, 35.8, 38., 4.5, 41.8, 42.8, 44.7, and 46. C,

4 VOL. 57, 1991 TEMPERATURE/GROWTH MODEL UTILITY ~- 1.o.1 Q. Cn I /Temperature (1/K) FIG. 1. Semilogarithmic plot of the specific growth rate versus the reciprocal of the absolute temperature for E. coli NC3 showing the result of the two-step method for fitting the six-parameter master reaction model (dashed line). Starting with the cutoff point, the high-temperature data were used to fit the four-parameter master reaction model (solid line). All data were used to fit the sixparameter master reaction model in the second step (see text). The error bars indicate the 95% confidence interval. ra ID ~ D :- wl 5 C_ c) : DD ' E EL U, ) ~ ~ ~ a a B o~~~~~~i oo -j (-A It l+ l+ l+1l+ C) O -S ffi. lp. 1s) O 4 % b~~~oo N 6 3 CA. 32 a U) <D o ~ whereas those used for growth in complex medium were 2, 25.1, 3., 35.5, 35.8, 38., 38.7, 39.5, 4.8, 42.8, 46., 47.54, and 48 C. The coccobacillus strain NA17 was grown in 2 liters of medium in a laboratory-scale bioreactor (Bioengineering AG, Wald, Switzerland) operating with an impeller speed of 8 rpm and an air flow rate of 35 liters/h and with the ph controlled at 6.8 by addition of either a 1 M NaOH-KOH solution or a 1% (wt/wt) H3P4 solution. The temperatures used for growth were 35.5, 4., 45., 5., 55., 57., 59., and 6. C. Growth was determined from measurements of the optical density at 546 nm. All experiments were carried out in duplicate, and datum points for calculated specific growth rates represent mean values. RESULTS U, \ o o o It o o I+ I+ -l -4x I+ I+I+ oa LAL 1+ I+ I+ 'IO % % *-.-. I+ I+1I+ o %o 'Il p ton I-I Hi.A r, P- w r ' x ro I+ co C % C~ O n. X (e Master reaction model. Preliminary fits of equation 6 to the experimental data by using the Marquardt-Levenberg algorithm resulted in unreliable parameter estimates. This behavior was largely due to the fact that in certain cases the x2 function (likelihood surface) has several local minima, thereby allowing good fits to be obtained in different sectors of parameter space by judicious choice of starting values for the individual parameters and resulting in biologically meaningless best-fit parameters (data not shown). However, when only high-temperature inactivation was considered by using a four-parameter model (equation 7), good stability was indicated and the model was independent of the starting values for the parameters over an acceptable range. Therefore, these initial problems could be resolved by developing a technique which involves two consecutive nonlinear Marquardt-Levenberg fits. The first step in this procedure involves fitting the four-parameter model to a partial data set. The cutoff point with respect to the data is determined by inspection of a semilogarithmic plot of the maximum specific growth rate versus the reciprocal of the absolute temperature, as shown in Fig. 1 for data for E. coli NC3. It is given by the point at which the descending branch of log,u as a function of 1T starts to deviate significantly from linear behavior. By ignoring data beyond the cutoff, i.e., the sub-arrhenius range, best estimates for r(25c), AHl, TO.5H, and AlH can be obtained. In the second step, the six- r F I+ It (A > oo N l_1 LA I+ I o) % 11A 4 tx.n GDe. C; p GD q p O= E

5 266 HEITZER ET AL. APPL. ENVIRON. MICROBIOL (D 1.5- ) 1. cm ) C.5 U) v.- CU1.6 J-c 2.4 cn t) cb v FIG. 2. Experimental data and fitted curves for the master reaction model. Solid lines represent the fits after the first step, and dashed lines represent the fits after the second step. (A) K. pneumoniae NCIB 418 grown in glucose mineral salts medium () and in complex medium (); (B) E. coli NC3 grown in complex medium; (C) NA17 grown in a mineral salt medium with ethanol as the carbon energy source; (D) Bacillus sp. strain NCIB grown in a mineral salt medium with methanol as the carbon energy source. The error bars indicate the 95% confidence interval. parameter model (equation 6) is fitted to the whole data set, maintaining the values fixed for the four parameters obtained in the first step, in order to obtain best estimates for both A'L and TO.SL. By using this approach, the fits obtained for the six-parameter model converged appropriately and were independent of the starting values chosen for the parameters, and best-fit estimates of the parameters remained within the biologically meaningful range. Estimated model parameters for the five data sets considered are given in Table 1; in Fig. 2 the experimental data are compared with the best-fit curves generated. For K. pneumoniae NCIB 418 (Fig. 2A), the higher maximum specific growth rates observed in a complex medium compared with those observed in a mineral medium are clearly evident, as are the different boundaries of both the subranges and the overall growth-permissible temperature range. With this bacterium growing on both media, comparison of the experimental data with the four- and six-parameter model curves indicates correspondence for the Arrhenius, optimum, and the first part of the superoptimum temperature ranges. In addition, the six-parameter model adequately describes the sub-arrhenius range. However, at the upper end of the superoptimum temperature range for growth, both models deviate markedly from the experimental data in that the asymptotic approaches of the model curves to the abscissa are insufficiently rapid. Comparison of the other data sets, i.e., those for E. coli NC3 (Fig. 2B), the thermotolerant coccobacillus NA17 (Fig. 2C), and the thermotolerant Bacillus sp. strain NCIB (Fig. 2D), with the respective model curves gives essentially similar results. This is because the model only asymptotically approaches a growth rate of zero. As can be seen in Table 1, the number of degrees of freedom for the fitted data sets significantly affects the relative magnitude of the standard deviations of the parameter estimates. In addition, the spreading of the individual experimental datum points can impact strongly on the standard deviations, as indicated for TOSL and AHL values for K. pneumoniae, strain NA17 and Bacillus sp. strain NCIB In contrast, the standard deviations obtained for the above-mentioned parameters of E. coli were significantly smaller, as one would expect, as a result of the larger

6 VOL. 57, 1991 TEMPERATURE/GROWTH MODEL UTILITY 2661 O~ / FIG. 3. Experimental data for K. pneumoniae in glucose mineral salts NCIB 418 grown identical even though their damage and repair parameters medium and fitted curve! s for the damage/ are markedly different. Furthermore, the values for KO and by changing the repair model. The different curves were obtained initial parameter values for the fitting procedure (T bars indicate the 95% confidence interval. number of datum points at lower temperatlures. judged from the The high x2 statistics, were in the same range. standard deviations of Al-PH for strain NA17 and spu Bacillus strain NCIB are due to a paucity of dat: um points in the superoptimum temperature range. tween the data sets and the models are most p fluctuations. For K. pneumoniae growing i dium, analysis of the residuals revealed that value is caused by a single point (p. = at 321 nonlinear problem. However, this algorithm provides no information about the standard deviations of the parameters. For K. pneumoniae NCIB 418 growing in mineral medium, the several best-fit curves obtained for the model with different initial conditions are compared with experimental data in Fig. 3, and the corresponding model parameters are given in Table 2. It can be seen that the final parameter values obtained were highly dependent on initial parameter values. By changing only one starting value for a single parameter, markedly different final parameter estimates resulted. Two essentially different-shaped curves resulted from the three different final parameter estimates (Fig. 3). L-_ * One curve gives a sharp peak at the optimum temperature for growth, whereas the other gives a much broader maximum. The two curves exhibiting sharp peaks are almost EA remain essentially constant, but are totally different from the values applicable to the curve with the broad maximum. Even so, the quality of fit between the different-shaped curves resulting from the model and the experimental data, The comparison of experimental data with results from the damage/repair model for K. pneumoniae NCIB 418 growing in complex medium, for E. coli NC3, for the coccobacillus e weighted fits strain NA17, and for Bacillus sp. strain NCIB is Analysis of x2 values obtained from th indicates that for K. pneumoniae NCIB 418 growing in shown in Fig. 4, and the corresponding model parameters mineral medium, for the coccobacillus NA174, and for Bacil- are given in Table 3. In contrast to the results obtained for lus sp. strain NCIB 12522, the apparent di: screpancies be- the master reaction model, the maximum growth temperasrobably chance n obablex chce- ture can be described by the damage/repair model. Model curves (Fig. 4A to C) decline precipitously to zero when the the elevated 2 damage rate equals the maximum possible repair rate. Howe K) and canebx ever, in the model for Bacillus sp. strain NCIB (Fig. It theauppernend 4D), judged as a problem of the model approach a this requirement is not fulfilled, and since the maximum upper End trthe of the superoptimum temperature range, whc co value of the dimensionless repair rate ereas for E. coli exceeds unity and the maximum damage rate cannot exceed unity, the damage rate NC3, the x2 value suggests that the stanidard deviation. hdard will never equal the repair rate, so that the curve will never assumed for the maximum specific growtl rates was an intersect the abscissa. This will always be the when the underestimate. best estimate for Damage/repair model. For the damage/repaiir model, equa- 12 is either smaller than or equal to the reciprocal of e (Euler's number; 1/2.718), as tions 8 to 1 and the specified boundary conditions define is evident from a function that cannot be differentiated at the temperature at equation 13. which V(T) = Pf3. In addition, the function i s discontinuous Square root model. To obtain best fits for the square root model, we used the at the maximum temperature for growth, wi weighted Marquard-Levenberg algorithm. As with the four-parameter master reaction model, iich is given by fits converged properly and final parameter estimates were 3- ln (e32-1) independent initially 1S K a Tmax = ] (13) (13) range. a2 Comparisons between the five sets of experimental data In this case, the Marquardt-Levenberg algo: rithm could not considered and model curves are shown in Fig. 5, and be applied, but a polytope algorithm, which dloes not assume corresponding model parameters are given in Table 4. Anal- from a par- ysis of the x2 values shows that discrepancies between the smoothness, showed satisfactory convergenice ticular set of starting values and was therefo)re used for this model and the experimental data are most probably caused TABLE 2. Dependence of the damage/repair model on the initial parameter estimates K. pneumoniae Best-fit model parameterb Statistical parameter NCIB 418 grownt pt Tmax inglucose(k () mediuma a2 (K-) a Ko In (1/h) EA (kj mo-1) (K) (K) x2 v Q 1.32 (1.) 55.9 (62.).39 (.5) (2.) (2.) (6.) x 1-6 (-.----) 1.22 (1.2) (62.).59 (.5) (2.) (2.) (6.) x 1-4 ( ) (2.) (62.) 2.53 (.5) 7.46 (2.) (2.) (6.) x 1-4 a The line styles indicated correspond to the curves in Fig Values of the best-fit model parameters (cs2, 3, 132' 133, Ko, and EA), of the calculated optimum and maximum growth temperatures (Tp, and Tmax), and of the associated statistical parameters (x2, v, and Q) are given for K. pneumoniae NCIB 418. The initial values of the parameters for each fit are given in parentheses.

7 2662 HEITZER ET AL. APPL. ENVIRON. MICROBIOL C C T- cm n ck 1. - DISCUSSION In analyses of microbial growth data, problems arising from error propagation, the analysis of errors in the estimated parameters, correlation of parameters, and consider-.5- '.' a) co ) cd.8 - a).6-4- L cn).2 ' (M v FIG. 4. Experimental data and fitted curves for the damage/repair model. (A) K. pneumoniae NCIB 418 grown in complex medium; (B) E. coli NC3 grown in complex medium; (C) NA17 grown in a mineral salt medium with ethanol as the carbon energy source; (D) Bacillus sp. strain NCIB grown in a mineral salt medium with methanol as the carbon energy source. The error bars indicate the 95% confidence interval. by chance fluctuations. The model, as used here (equation 12), is able to describe the temperature behavior of all data sets adequately. For the coccobacillus strain NA17, an overestimation of the standard deviation for the data set is indicated, as Q is close to unity. It can be seen from Table 4 that the standard deviations associated with the minimum temperatures for growth are significantly larger than those associated with the maximum temperatures for growth for all five data sets. In addition to other factors, a paucity of datum points in the sub-arrhenius range contributes to this finding. TABLE 3. Values of the best-fit model parameters, of the calculated optimum and maximum growth temperatures, and of the associated statistical parameters obtained from a nonlinear fit to the damage/repair model Best-fit model parameter Statistical parameter Bacterial strain T.Pt Tmax a2 (K-1) a Ko ln(1/h) EA (kj (K) (K) X2 v mol-')x Q K. pneumoniae NCIB 418 grown in complex medium E. coli NC x 1-3 Strain NA Bacillus sp. strain NCIB x jq-3

8 VOL. 57, 1991 TEMPERATURE/GROWTH MODEL UTILITY J- v cs ) n.5 cn.5 m Z w-.6 } C,,.4 2 cn FIG. 5. Experimental data and fitted curves for the square root model. (A) K. pneumoniae NCIB 418 grown in glucose mineral salts medium () and in complex medium (); (B) E. coli NC3 grown in complex medium; (C) NA17 grown in a mineral salt medium with ethanol as the carbon energy source; (D) Bacillus sp. strain NCIB grown in a mineral salt medium with methanol as the carbon energy source. The error bars indicate the 95% confidence interval. ations of the goodness of fit are frequently neglected (1, 15, 16, 25). However, for the models under consideration in this study, nonlinear least-square techniques have been used as maximum likelihood estimators, and the results clearly illustrate that mere phenomenological inspection of a fitted model curve and its comparison with the experimental data are not sufficient to evaluate the quality of the fit and the utility of a mathematical model. A most important question that should be addressed before using any mathematical model to describe a particular biological response is the objective. In general, three main areas of interest can be identified for the specific-growthrate-versus-temperature relationship. (i) A particular model might be used to predict the specific growth rate of a culture beyond the experimentally investigated temperature range. (ii) The objective is to determine characteristic parameters for comparative reasons, which implies simple interpretation and problem-free determination of the parameters. (iii) The TABLE 4. Values of the best-fit model parameters, of the calculated optimum growth temperature, and of the associated statistical parameters obtained from a nonlinear least-square fit to the square root model Bacterial strain Best-fit model parameter + s TP, Statistical parameter K 2 b(k- t-5) c(k1) Tmin (K) Tmax (K) (K) X2 v Q K. pneumoniae NCIB 418 Glucose medium.39 ± ± ± ± Complex medium.44 ± ± ± ± E. coli NC3.413 ± ± ± ± Strain NA ± ± ± ± Bacillus sp. strain NCIB ± ± ± ±

9 2664 HEITZER ET AL. model is expected to provide insight into fundamental physiological processes that might control the growth rate response. Considering the three models discussed with respect to these points, it is evident that all the models investigated are able to describe the specific-growth-rate-versus-temperature relationship over almost the entire growth-permissible temperature range. However, when using the Marquardt-Levenberg algorithm, acceptable fits could be obtained only for the master reaction and square root models. In the damage/ repair model, the model function has locations of nondifferentiability and of noncontinuity and, further, requires that the activity state variable, V, be evaluated numerically, so that smoothness of the function does not result. These problems were overcome by the use of the polytope algorithm. However, the resulting fits showed strong dependence on the starting values of the various parameters. The data of Soeder et al. (34) were also fitted to the model function but failed to generate the parameter values given for the same data (14), although fits were obtained with a smaller sum of the squared deviations, indicating that the values reported (14) did not correspond to a global minimum. Apparently the x2 function possesses several six-dimensional minima in a flat valley of complicated topology such that different starting values for the parameters lead to different local minima that are "virtually as good" as a global minimum (9). The model is unsuited for the data sets considered, indicating that it is overparameterized. This does not necessarily mean that the model is inapplicable under all circumstances, but simply that the available data are inadequate for estimating all the postulated parameters. Even so, compared with the other models evaluated, the damage/repair model has the least practical value, because the parameters are neither easily estimated nor interpretable. If the model can be successfully freed of its intrinsic mathematical inconsistencies, it might be of interest because kinetically structured approach to temperature inacti- of its vation. A similar situation to that discussed for the damage/repair model applies to the original master reaction model (equation 5), but it has been shown (32) that by appropriate reparameterization, proper convergence could be obtained for the four-parameter model (equation 7). Direct fits with the six-parameter model (equation 6) exhibit a dependence on starting values for the parameters, and, especially with small data sets, minima with biologically meaningless parameter estimates result. The two-step method proposed herein eliminates most of these problems. It has been suggested (1) that logarithmic transformation of the master reaction model might improve its numerical stability and partially reduce its dependence on the starting values chosen for the parameters, but unfortunately neither discussion of error nor indication that weighted least-square techniques were used were provided. As this is a prerequisite for obtaining statistically interpretable parameter estimates, this proposal could not be pursued further. Problems associated with the interpretation of the model parameters in the master reaction model are somewhat similar to the situation for Monod growth kinetics. Monod growth kinetics use KS values that are mathematically analogous to Michaelis-Menten KMvalues, but can no longer be interpreted as a ratio of rate constants. The same is true for the thermodynamic constants appearing in the master reaction model. They can no longer be interpreted as true thermodynamic properties, unless a single growth-rate-determining reaction can be identified. As this is usually not the APPL. ENVIRON. MICROBIOL. case, these model parameters must be considered to be experimentally evaluated specific strain constants under defined nutrient conditions. Nevertheless, they can be of value for comparing and discussing the temperature behavior of different cultures, very much as Ks values can be used to compare and discuss growth behavior under changing substrate concentrations. It is interesting that the thermodynamic parameters AH, AHL, and AHH are of the same order of magnitude for all the strains evaluated. The AHH values are also comparable to the values for different proteins (27). This is an indication that the theoretical approach used in formulating the model maintains a degree of biological reality; e.g., the macroscopic behavior of the bulk enzyme system used for growth is similar in its temperature response to that for a single enzyme reaction. The situation is different for the square root model. Although the model defines cardinal temperatures (Tmin and Tmax), at least Tmin cannot be interpreted as the minimum temperature of growth (24) but must be considered a conceptual temperature. However, it has been noted that although the actual minimum temperature for growth may occur several degrees above Tmin, a lot of growth data indicate that the model remains valid to the point at which growth ceases (24). There is, as yet, no physiological explanation for the square root response of bacterial growth rate to temperature, so that the remaining parameters, b and c, constitute mere regression coefficients. Tables 1 and 4 show that the precision of the estimated parameters strongly depends on the number of degrees of freedom and the spread of the datum points. As can be seen for E. coli NC3, the precision dramatically increased with increasing degrees of freedom. Scoring the three models that have been evaluated in this paper in accordance with the criteria outlined above for judging the utility of unstructured phenomenological models indicates that the square root model and the master reaction model are placed first and the damage/repair model is placed third, even though the square root model is devoid of any conceptual basis. In general, the quality of parameter estimation and the predictive value of models can be improved by more extensive experimental data with a reasonable spread over the entire growth-permissible temperature range. Furthermore, fitting problems due to the intrinsic mathematical structure of a model can be improved by appropriate reparameterization or, as has been shown here, by using a two-step procedure with partial data sets. REFERENCES 1. Adair, C., D. C. Kilsby, and P. T. Whitall Comparison of the Schoolfield (nonlinear Arrhenius) model and the square root model for predicting bacterial growth in foods. Food Microbiol. 6: Al-Awadhi, N The characterization and physiology of some thermotolerant and thermophilic solvent-utilizing bacteria. Doc. Diss. ETH no Eidenossische Technische Hochschule, Zurich, Switzerland. 3. Al-Awadhi, N., T. Egli, and G. Hamer Growth characteristics of a thermotolerant methylotrophic Bacillus sp. (NCIB 12522) in batch culture. Appi. Microbiol. Biotechnol. 29: Al-Awadhi, N., G. Hamer, and T. Egli Effects of temperature on the growth rate of the thermotolerant methylotrophic Bacillus sp. NCIB in chemostat culture. Acta Biotechnol. 1: Arrhenius, S Immunochemie. Ergeb. Physiol. 7: Belehradec, J Influence of temperature on biological processes. Nature (London) 118:

10 VOL. 57, Brandts, J. F Heat effects on proteins and enzymes, p In A. H. Rose (ed.), Thermobiology. Academic Press, Inc. (London), Ltd., London. 8. Crozier, W. J On biological oxidations as function of temperature. J. Gen. Physiol. 7: Draper, N., and H. Smith Applied regression analysis, 2nd ed. John Wiley & Sons, Inc., New York. 1. Eisenberg, D., and D. Crothers Physical chemistry with applications to the life sciences. The Benjamin/Cummings Publishing Co., Menlo Park, Calif. 11. Esener, A. A., J. A. Roels, and N. W. F. Kossen The influence of temperature on the maximum specific growth rate of Klebsiella pneumoniae. Biotechnol. Bioeng. 23: Evans, C. G. T., D. Herbert, and D. W. Tempest The continuous cultivation of microorganisms, p In J. R. Norris and D. W. Ribbons (ed.), Methods in microbiology, vol. 2. Academic Press, Inc. (London), Ltd., London. 13. Eyring, H., and D. W. Urry Thermodynamics and chemical kinetics, p In T. H. Waterman and H. J. Morowitz (ed.), Theoretical and mathematical biology. Blaisdell Publishing Co., New York. 14. Fiolitakis, E., U. Grobbelaar, E. Hegewald, and C. J. Soeder Deterministic interpretation of the temperature response of microbial growth. Biotechnol. Bioeng. 3: Gibson, A. M., N. Bratchell, and T. A. Roberts The effect of sodium chloride and temperature on the rate and extent of growth of Clostridium botulinum type A in pasteurized pork slurry. J. Appl. Bacteriol. 62: Gill, C. O., and D. M. Phillips The effect of media composition on the relationship between temperature and growth rate of Escherichia coli. Food Microbiol. 2: Hamer, G., and A. Heitzer Fluctuating environmental conditions in scaled-up bioreactors-heating and cooling effects. Ann. N.Y. Acad. Sci. 589: Heitzer, A Kinetic and physiological aspects of bacterial growth at superoptimum temperatures. Doc. Diss. ETH no Eidgenossische Technische Hochschule, Zurich, Switzerland. 19. Herendeen, S. L., R. A. Van Bogelen, and F. C. Neidhardt Levels of major proteins of Escherichia coli during growth at different temperatures. J. Bacteriol. 139: Ingraham, J. L Effect of temperature, ph, water activity, and pressure on growth, p In F. C. Neidhardt, J. L. Ingraham, K. B. Low, B. Magasanik, M. Schaechter, and H. E. Umbarger (ed.), Escherichia coli and Salmonella typhimurium: cellular and molecular biology. American Society for Microbiology, Washington, D.C. 21. Johnson, F. H., and I. Lewin The growth rate of E. coli in TEMPERATURE/GROWTH MODEL UTILITY 2665 relation to temperature, quinine and coenzyme. J. Cell. Comp. Physiol. 28: Kohler, H.-P. E., A. Heitzer, and G. Hamer Improved unstructured model describing temperature dependence of bacterial maximum specific growth rates, p In H. Verachtert and W. Verstraete (ed.), International Symposium on Environmental Biotechnology. Royal Flemish Society of Engineers, Ostend, Belgium. 23. McMeekin, T. A., R. E. Chandler, P. E. Doe, C. D. Garland, J. Olley, S. Putros, and D. A. Ratkowsky Model for combined effect of temperature and salt concentration/water activity on the growth rate of Staphylococcus xylosus. J. Appl. Bacteriol. 62: McMeekin, T. A., J. Olley, and D. A. Ratkowsky Temperature effects on bacterial growth rates, p In M. Bazin and J. Prosser (ed.), Mathematical models in microbiology. CRC Press, Boca Raton, Fla. 25. Mohr, P. W., and S. Krawiec Temperature characteristics and Arrhenius plots for nominal psychrophiles, mesophiles and thermophiles. J. Gen. Microbiol. 121: Monod, J Recherches sur la croissance des cultures bacteriennes. Hermann, Paris. 27. Morowitz, H. J Energy flow in biology, p Academic Press, Inc., New York. 28. Neidhardt, F. C., J. L. Ingraham, and M. Schaechter Physiology of the bacterial cell. Sinauer Associates, Inc., Sunderland, Mass. 29. Press, W. A., B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling Numerical recipes, the art of scientific computing. Cambridge University Press, Cambridge. 3. Ratkowsky, D. A., R. K. Lowry, T. A. McMeekin, A. N. Stokes, and R. E. Chandler Model for bacterial culture growth rate throughout the entire biokinetic temperature range. J. Bacteriol. 154: Ratkowsky, D. A., J. Olley, T. A. McMeekin, and A. Ball Relationship between temperature and growth rate of bacterial cultures. J. Bacteriol. 149: Schoolfield, R. M., P. J. H. Sharpe, and C. E. Magnuson Nonlinear regression of biological temperature-dependent rate models based on absolute reaction-rate theory. J. Theor. Biol. 88: Sharpe, P. J. H., and D. W. DeMichele Reaction kinetics of poikilotherm development. J. Theor. Biol. 64: Soeder, C. J., E. Hegewald, E. Fiolitakis, and J. U. Grobbelaar Temperature dependence of population growth in a green microalga: thermodynamic characteristics of growth intensity and influence of cell concentration. Z. Naturforsch. Teil C 4: